Let the maximum length sequence $$s_n=(-1)^{Tr(\beta \alpha^n)}$$have period $2^m-1,$ and be nontrivial with $\beta\neq0.$ Your sum is $$\Gamma(m,N)=\sum_{n=1}^N s_n \exp(2 \pi i n/N).$$
If $\beta=0,$ you have the standard unmodulated linear exponential sum with the bound $$\min\left\{N,\frac{1}{|\sin \pi/N|}\right\}. $$
If $N\leq m,$ all possible $N-$tuples in $\{\pm 1\}^N$ are taken on by $$(s_n,\ldots,s_{n+N-1})$$ so a better bound is not possible.
Edit: In the intermediate range $m<N\leq 2^m-1,$ a bound along the lines suggested by @FelipeVoloch is indeed possible.
From Theorem 8.78 in the book Lidl and Niederreiter, Finite Fields (Encyclopedia of Mathematics and its Applications, we have $$ |\Gamma(m,N)|\leq \sqrt{2^m}, $$ which is nontrivial if $N>\sqrt{2^m}$.